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arXiv:2605.31248

Trainable Quantum Spectral Models for Partial Differential Equations

Gabriel Mejia, Eileen Kuehn, Melvin Strobl, Achim Streit

quant-phSubmitted May 29, 2026• Announced Jun 1, 2026• 0 citations
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Abstract

This work studies trainable quantum spectral models (QSMs) for solving linear partial differential equations (PDEs). Instead of learning solutions directly in physical space, QSMs learn the inverse differential operator in a spectral representation, embedding prior knowledge of the equation's natural basis. We systematically study the expressibility and trainability of several QSM architectures, ranging from near-diagonal to fully parameterized unitaries. In particular, we introduce a family of richer spectral models that interpolate between purely diagonal operators and fully mixing unitaries through a parameterized mixer controlled by $ε$. Our results reveal an intermediate regime, typically around $ε\approx 0.5$, where models achieve the best tradeoff between expressibility and trainability. Beyond this threshold, increased circuit complexity degrades convergence without improving accuracy. Among the architectures considered, models inspired by the inverse step of the Harrow-Hassidim-Lloyd (HHL) algorithm achieve the fastest training convergence while maintaining high solution fidelity. Numerical experiments on the (variable-coefficient) Poisson and Helmholtz equations show that trainable operations in the spectral basis outperform standard variational quantum circuits acting directly in the computational basis. These advantages appear through faster convergence, more stable gradients, and more accurate recovery of the reference solution spectrum, particularly through stronger suppression of spurious high-frequency components, even when the operator is not exactly diagonal in the chosen spectral basis. Our results identify operator-aware spectral representations as a promising route toward trainable and physically grounded quantum methods for scientific computing.

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This work studies trainable quantum spectral models (QSMs) for solving linear partial differential equations (PDEs). Instead of learning solutions directly in physical space, QSMs learn the inverse differential operator in a spectral representation, embedding prior knowledge of the equation's natural basis.
We systematically study the expressibility and trainability of several QSM architectures, ranging from near-diagonal to fully parameterized unitaries. In particular, we introduce a family of richer spectral models that interpolate between purely diagonal operators and fully mixing unitaries through a parameterized mixer controlled by εεε. Our results reveal an intermediate regime, typically around ε≈0.5ε\approx 0.5ε≈0.5, where models achieve the best tradeoff between expressibility and trainability. Beyond this threshold, increased circuit complexity degrades convergence without improving accuracy.
Among the architectures considered, models inspired by the inverse step of the Harrow-Hassidim-Lloyd (HHL) algorithm achieve the fastest training convergence while maintaining high solution fidelity. Numerical experiments on the (variable-coefficient) Poisson and Helmholtz equations show that trainable operations in the spectral basis outperform standard variational quantum circuits acting directly in the computational basis. These advantages appear through faster convergence, more stable gradients, and more accurate recovery of the reference solution spectrum, particularly through stronger suppression of spurious high-frequency components, even when the operator is not exactly diagonal in the chosen spectral basis. Our results identify operator-aware spectral representations as a promising route toward trainable and physically grounded quantum methods for scientific computing.
Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.
Quantum methods for solving partial differential equations (PDEs) generally fall into a few general categories, each leveraging different quantum mechanisms to achieve potential speedups. One well-known approach is the quantum linear system solver, epitomized by the Harrow-Hassidim-Lloyd (HHL) algorithm, which uses phase estimation and eigenvalue inversion to solve large, sparse linear systems arising from discretized linear PDEs [1]. This method can yield exponential improvements over classical solvers under certain conditions (mostly when the matrix is well-conditioned and sparse). More recent developments, such as Quantum Singular Value Transformation (QSVT), provide a framework for matrix inversion in a more flexible way, often improving on the cost and error dependence of earlier solvers [2]. Time-dependent PDEs can be tackled via Hamiltonian simulations. The Trotter-Suzuki method decomposes the exponential of a Hamiltonian into products of exponentials of simpler sub-Hamiltonians, allowing a piecewise approximation of continuous dynamics [3]. Continuous-time quantum walks, which treat a PDE operator as a Hamiltonian that governs the walk, similarly simulate processes like diffusion or wave propagation on discretized grids or graphs [4]. Beyond direct simulation, variational quantum algorithms have emerged as a promising route in the noisy intermediate-scale quantum era. These approaches encode the PDE solution as a parameterized quantum state and optimize the parameters via classical feedback, thereby minimizing a cost function tied to the PDE residual or energy functional [5]. Quantum annealing or adiabatic methods, meanwhile, can solve optimization problems derived from PDE constraints by adiabatically evolving a simple initial Hamiltonian to the final problem Hamiltonian and staying in the system's ground state [3]. In our study, we implement a new parameterized quantum model, specifically designed to solve large linear systems on quantum hardware. This model is less deep and computationally faster than the proposed one in [5], reducing the complexity and speeding up convergence times. Such improvements are particularly valuable in cases where classical algorithms may fail due to the problem's size or complexity. Ongoing research is focusing on generalized quantum models that can be adapted to practical PDE problems for solving large electromagnetic problems where classical computers can fail.
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity p o l y ( 1 / ϵ ) , where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be p o l y ( d , log ⁡ ( 1 / ϵ ) ) , where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.
We develop and analyze generative models based on quantum solvers of stochastic differential equations. Unlike the common approach based on layered variational quantum circuits with parameterized rotations, we parameterize the potential of a Langevin process. The associated Fokker-Planck (FP) equations are Hermitianized and solved using a linear combination of cosine operators implemented through quantum singular value transformation. By encoding the square root of the stationary density, the model enables sampling through computational-basis measurements and is efficiently controlled by coefficients of the block-encoded diagonal potential. We train the proposed model to match empirical distributions within the family described by FP equations, demonstrating parameter recovery for multivariate probability distributions. Crucially, the model remains trainable at scale. Sample-efficient and convex optimization for in-distribution targets arises from a physics-informed inductive bias and holds regardless of the dimensionality of the problem. We provide circuit constructions and efficient block encodings of the main components of the algorithm and analyze resource requirements. Our approach frames quantum generative learning as quantum model discovery and provides a foundation for extending this framework to high-dimensional, strongly correlated scientific data.
Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) efficiently. Recently, variational approaches to solving non-linear PDEs also with near-term quantum devices were proposed. One of the most promising general approaches is based on recent developments in the field of scientific machine learning for solving PDEs. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks, including the discovery of differential equations from a dataset of measurements. We use differentiable quantum circuits (DQCs) to solve equations parameterized by a library of operators, and perform regression on a combination of data and equations. Our results show a promising path to Quantum Model Discovery (QMoD), on the interface between classical and quantum machine learning approaches. We demonstrate successful parameter inference and equation discovery using QMoD on different systems including a second-order, ordinary differential equation and a non-linear, partial differential equation.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.